A unified approach to the spectral radius, connectivity and edge-connectivity of graphs

For two integers r ≥ 2 and h ≥ 0 , the h-extra r-component connectivity κ r h ( G ) of a graph G is defined as the minimum size of a subset S of vertices whose removal disconnects G , such that there are at least r connected components in G − S and each component has at least h + 1 vertices. Denote by G n , δ κ r h the set of n -vertex graphs with h -extra r -component connectivity κ r h and minimum degree δ . The following problem concerning spectral radius was proposed by Brualdi and Solheid (1986) [2] : Given a set of graphs S , find an upper bound for the spectral radius of graphs in S and characterize the graphs in which the maximum spectral radius is attained. We study this question for S = G n , δ κ r h where r ≥ 2 and h ≥ 0 . Fan, Gu and Lin (2024) [7] answered the question for r ≥ 2 and h = 0 . In this paper, we solve this problem completely for r ≥ 2 and h ≥ 1 . Moreover, we also investigate analogous problems for the edge version. This implies some previous results in connectivity and edge-connectivity.